Proposition 6

— Let T be a completely regular space.

a) Let f be a lower semi-continuous numerical function $\geqslant 0$ defined on T; then the function $\mu \mapsto |\mu|^*(f)$ is lower semi-continuous on $\mathscr{M}^b(T)$.

b) Let f be an upper semi-continuous bounded function defined on T; then the function $\mu \mapsto \mu(f)$ is upper semi-continuous on $\mathscr{M}_+^b(T)$.

For, one sees by Prop. 1 b) of No. 1 that $\mu \mapsto |\mu|^*(f)$ is the upper envelope of a family of functions of the form $\mu \mapsto |\mu(g)|$ with $g \in \mathscr{C}^b(T)$, hence continuous for the tight topology. This establishes a). To prove b), it suffices to choose a constant upper bound C for f, and to write $\mu(f) = \mu(C) - \mu(C - f)$; the function $\mu \mapsto \mu(C)$ is continuous, and the function $\mu \mapsto \mu(C - f)$ is lower semi-continuous on $\mathscr{M}_+^b(T)$ by the foregoing.